Developing the Next Generation Scalable Exascale Uncertainty Quantification Methods

Predictive modeling of multiscale and multiphysics systems requires accurate data-driven characterization of the input uncertainties and understanding how they propagate across scales and alter the final solution. We will address three major current limitations in modeling stochastic systems: (1) Most of current uncertainty quantification methods cannot detect and handle discontinuity in the parametric space, (2) Lack of scalable uncertainty quantification methods to take advantage of the future exascale computing facilities, and (3) Lack of efficient nonlinear manifold learning tools and model reduction methods for high-dimensional, stochastic multiscale systems. The first limitation is addressed with the development of a multi-output Gaussian process model. The second limitation is addressed with the construction of scalable multigrid methods. The last limitation is addressed with the diffusion map based nonlinear manifold tools for efficient model reduction of high-dimensional, stochastic multiscale systems. Computationally tractable low-complexity surrogate models, e.g. the multi-output Gaussian process model we developed, will play a significant role in the future in harnessing and controlling complex stochastic systems. Our integrated methodology involves concepts from diffusion map based manifold learning, multi-output Gaussian process model, scalable multigrid methods for high dimensional stochastic PDE systems and scalable parallel algorithms. Numerical results demonstrated the capability and efficiency of the developed scalable uncertainty quantification methods.